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Boundary Value Problems, Integral Equations and Related Problems Downloaded from www.worldscientific.com
by NATIONAL UNIVERSITY OF SINGAPORE on 10/25/17. For personal use only.
MODE I CRACK PROBLEM FOR A
FUNCTIONALLY GRADED ORTHOTROPIC
COATING-SUBSTRATE STRUCTURE1
LI-FANG GUOa,b , XING LIc and SHENG-HU DINGc
a
Higher Health Vocational and Technical College,
Ningxia Medical University, Yinchuan 750021, China.
b
Department of Mathematics, Shanghai Jiaotong University,
Shanghai 200240, China.
E-mail: glf1998@163.com
c
School of Mathematics and Computer Science, Ningxia University,
Yinchuan 750021, China
In this paper, the Fourier integral transform-singular integral equation method
is presented for the Mode I crack problem of the functionally graded orthotropic coating-substrate structure. The elastic property of the material
is assumed vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the boundaries of the
strip. Numerical examples are presented to illustrate the effects of the crack
length, the material nonhomogeneity and the thickness of coating on the stress
intensity factors.
Keywords: Functionally graded orthotropic material, coating-substrate structure, mode I crack problem, singular integral equation.
AMS No: 35J65, 35J55, 35J45.
1.
Introduction
The analysis of functionally graded materials has become a subject of increasing importance motivated by a number of potential benefits from the
use of such novel materials in a wide range of modern technological practices. The major advantages of the graded material, especially in elevated
temperature environments stem from the tailoring capability to produce
a gradual variation of its thermomechanical properties in the spatial domain. A Great efforts have been made to study the fracture behavior of
FGMs [1–5]. As it is reported in the literature [6–7], the graded materials
are rarely isotropic because of the nature of techniques used in fabricating them. Thus, it is necessary to consider the anisotropic character when
studying the failure behaviors of FGMs. Guo et al. [8] investigated the
mode I surface crack problem for an orthotropic graded strip. H. M. Xu
et al. [9] studied the problems of a power-law orthotropic and half-space
1 This research is supported by NSFC (10962008) and (51061015) and NSF of Ningxia
(NZ1001)
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290
Li-Fang Guo, Xing Li and Sheng-Hu Ding
functionally graded material (FGM) subjected to a line load. In the paper [10], asymptotic analysis coupled with Westergaard stress function approach is used to develop quasi-static stress fields for a crack oriented along
one of the principal axes of inhomogeneous orthotropic medium. Kim and
Paulino [11–12] examined mixed-mode stress intensity factors for cracks arbitrarily oriented in orthotropic FGMs using modified crack closure method
and the path-independent Jk∗ -integral, respectively.
Though there are lots of papers related to the crack problem of orthotropic functionally graded materials, very few papers on the plane crack
problem of a functionally graded strip with a crack perpendicular to the
boundary are published. It is very significant to study this kind of crack
problem, since the geometry can be used as an approximation to a number
of structural components and laboratory specimens, at the same time in
line with the result of the Kawasaki and Watanabe[13]’s experiments about
the thermal fracture behavior of metal/ceramic functionally graded materials(PSZ/IN 100 FGMS and PSZ/Inco 718 FGMs), the sequence of spalling
behavior was found to be: crack vertical to the sample surface formed during cooling, then transverse crack formed in graded layer during heating,
and subsequent growth of transverse cracks and their coalescence led eventually the ceramic coating to spall. Therefore, the surface crack problem is a
very important issue to be considered during the design of FGMs. In these
papers, the Mode I crack problem for a functionally graded orthotropic
coating-substrate structure will be presented and the results could be useful in the laboratory test and the design of the orthotropic functionally
graded materials .
2.
Formulation of the Problem
x
b
h
2a
FGM orthotropic
0
c
1
a
o
y
substrate
2
Figure 1:
The geometry of the Mode I crack problem for the functionally graded
orthotropic substrate-coating structure .
As shown in Fig.1: A functionally graded orthotropic strip with prop-
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Mode I Crack Problem for a Substrate-Coating Structure
291
erties varying in the x-direction bonded to a half infinite orthotropic elastic
substrate. The strip is infinite along the y-axis and has a thickness h along
x-axis. The principal direction of orthotropy are parallel and perpendicular
to the boundaries of the strip.
The material properties are defined as
c11 (x) = c110 eδx, c12 (x) = c120 eδx, c22 (x) = c220 eδx, c66 (x) = c660 eδx,
(1)
where c110 , c120 , c220 , c660 and δ are constants. c110 , c120 , c220 and c660 are
the material parameters of y = 0, δ is the gradient parameters of functionally graded material.
The general constitutive relation can be written as
σ1xx = c11 (x)
∂u
∂v
∂u
∂v
+c12 (x)
, σ1yy = c12 (x)
+c22 (x)
,
∂x
∂y
∂x
∂y
µ
¶
∂u ∂v
σ1xy = c66 (x)
+
.
∂y
∂x
(2)
The equilibrium equation in terms of the displacements can be given as:
∂σ1xx
∂σxy
∂σ1yy
∂σxy
+
= 0,
+
= 0,
∂x
∂y
∂y
∂x
(3)
make use of the equation Eqs. (1), (2) and (3), it can be obtained that
∂2u
∂2u
∂2v
+c
(x)
+
[c
(x)+c
(x)]
66
12
66
∂x2
∂y 2
∂x∂y
∂u
∂v
+c11 (x) δ +c12 (x) δ
= 0,
∂x
∂y
∂2v
∂2v
∂2u
c22 (x) 2 +c66 (x) 2 + [c12 (x) + c66 (x)]
∂y
∂x
∂x∂y
∂u
∂v
+c66 (x) δ +c66 (x) δ
= 0,
∂y
∂x
c11 (x)
(4)
if let the δ = 0, then the equations of the elastic substrate can be obtained.
The mixed boundary conditions of the problem in Fig.1 can be written
as
σ1xx (h, y) = 0, σ1xy (h, y) = 0, σ1xx (0, y) = σ2xx (0, y) ,
σ1xy (0, y) = σ2xy (0, y) , u1 (0, y) = u2 (0, y) , v1 (0, y) = v2 (0, y) ,
σ2xx (x, y) = 0, σ2xy (x, y) = 0, x → −∞,
σ1xy (x, 0) = 0, 0 < x < h, σ1yy (x, 0) = −σ0 (x) , a < x < b,
v1 (x, 0) = 0, 0 < x < a, b < x < h.
(5)
(6)
(7)
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292
Li-Fang Guo, Xing Li and Sheng-Hu Ding
By use of the Fourier transform method and thinking of the Eq. (6), the
following displacement forms can be obtained, for functionally graded coating:
Z ∞X
4
1
u1 =
E1j (s) A1j eλ1j (s)y−isx ds
2π −∞ j=3
Z
4
2 ∞X
+
E2j (α) A2j eλ2j (α)x cos αydα,
π 0 j=1
(8)
Z ∞X
4
1
λ1j (s)y−isx
v1 =
A1j e
ds
2π −∞ j=3
Z
4
2 ∞X
+
A2j eλ2j (α)x sin αydα,
π 0 j=1
for elastic substrate:
2
u2 =
π
2
v2 =
π
Z
2
∞X
0
Z
A3j (α) E3j eλ3j (α)x cos αydα,
j=1
2
X
(9)
∞
0
λ3j (α)x
A3j e
sin αydα,
j=1
where the coefficient Eij , λij (i = 1, 2, 3, j = 1, · · · , 4) are shown in Appendix A, Aij (i = 1, 2, 3, j = 1, · · · , 4) are unknown functions, which can be
solved by the boundary conditions. To obtain the integral equations, let’s
introduce the following auxiliary function
g (x) =
∂
v1 (x, 0) ,
∂x
(10)
and g (x) subjected to the following single-valuedness conditions
Z
b
g (x) dt = 0.
(11)
a
By using Eqs. (7), (10) and applying the Fourier transform to Eq. (10), it
can be obtained that
Z b
Z b
isu
A13 = q13 g (u) e du, A14 = q14 g (u) eisu du,
(12)
a
where q13 =
−i(c120 s2 +c220 λ214 )
c220 s(λ213 −λ214 )
a
, q14 =
i(c120 s2 +c220 λ213 )
c220 s(λ213 −λ214 )
.
293
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Mode I Crack Problem for a Substrate-Coating Structure
Then by residue theorem and the boundary conditions,
following linear algebraic equations
¯
¯¯
¯ B21 eλ21 h B22 eλ22 h B23 eλ23 h B24 eλ24 h 0
0 ¯¯ ¯¯ A21
¯
¯
¯¯
0 ¯ ¯ A22
¯ F21 eλ21 h F22 eλ22 h F23 eλ23 h F24 eλ24 h 0
¯
¯¯
¯ B21
B22
B23
B24 −B31 −B32 ¯¯ ¯¯ A23
¯
¯ F
F22
F23
F24
−F31 −F32 ¯¯ ¯¯ A24
21
¯
¯
¯¯
E22
E23
E24 −E31 −E32 ¯ ¯ A31
¯ E21
¯
¯¯
¯
1
1
1
1
1
1 ¯ ¯ A32
where
we can get the
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯=¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯ ¯
¯
R1 ¯¯
R2 ¯¯
¯
R3 ¯¯
, (13)
R4 ¯¯
¯
R5 ¯
¯
R6 ¯
¢
−c2120 + c110 c220 α2 λ23
R1 (u) =
e−λ23 (u−h)
c110 (λ23 − λ21 ) (λ23 − λ22 ) (λ23 − λ24 )
¡ 2
¢
−c120 + c110 c220 α2 λ24
e−λ24 (u−h) ,
+
c110 (λ24 − λ21 ) (λ24 − λ22 ) (λ24 − λ23 )
¡ 2
¢
−c120 + c110 c220 αλ23 (λ23 + δ)
R2 (u) = −
eλ23 (h−u)
c110 (λ23 − λ21 ) (λ23 − λ22 ) (λ23 − λ24 )
R3 (u) =
R4 (u) =
R5 (u) =
R6 (u) =
¡
+c110 (λ24 − λ21 ) (λ24 − λ22 ) (λ24 − λ23 )eλ23 (h−u) ,
¡
¢
− −c2120 + c110 c220 α2 λ21
e−λ21 u
c110 (λ21 − λ22 ) (λ21 − λ23 ) (λ21 − λ24 )
¡
¢
− −c2120 + c110 c220 α2 λ22
+
e−λ22 u ,
c110 (λ22 − λ21 ) (λ22 − λ23 ) (λ22 − λ24 )
¡ 2
¢
−c120 + c110 c220 αλ21 (λ21 + δ)
e−λ21 u
c110 (λ21 − λ22 ) (λ21 − λ23 ) (λ21 − λ24 )
¡ 2
¢
−c120 + c110 c220 αλ22 (λ22 + δ)
+
e−λ22 u ,
c110 (λ22 − λ22 ) (λ22 − λ23 ) (λ22 − λ24 )
1 h c120 λ221 + 2δc120 λ21 + c120 δ 2 + α2 c220 −λ21 u
e
c110 (λ21 − λ22 ) (λ21 − λ23 ) (λ21 − λ24 )
c120 λ222 + 2δc120 λ22 + c120 δ 2 + α2 c220 −λ22 u i
+
e
,
(λ22 − λ21 ) (λ22 − λ23 ) (λ22 − λ24 )
¡ 2
¢ 2
−α −c120 + c110 c220 − c120 c660 λ21
e−λ21 u
c110 c660 λ21 (λ21 − λ22 ) (λ21 − λ23 ) (λ21 − λ24 )
¡
¢
¡
¢
αδλ21 c2120 −c110 c220 +2c120 c660 −αc660 δ 2 c120 +c220 α2 −λ21 u
+
e
c110 c660 λ21 (λ21 − λ22 ) (λ21 + λ23 ) − (λ21 − λ24 )
¡ 2
¢ 2
−α −c120 + c110 c220 − c120 c660 λ22
e−λ22 u
+
c110 c660 λ22 (λ22 − λ21 ) (λ22 − λ23 ) (λ22 − λ24 )
294
Li-Fang Guo, Xing Li and Sheng-Hu Ding
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¡
¢
¡
¢
αδλ22 c2120 −c110 c220 +2c120 c660 −αc660 δ 2 c120 +c220 α2 −λ22 u
+
e
c110 c660 λ22 (λ22 − λ21 ) (λ22 − λ23 ) (λ22 − λ24 )
¢
¡ 2
α δ c120 + c220 α2
+
.
c110 λ21 λ22 λ23 λ24
During the process of obtaining Eq. (13), the integral identities A6 [14]
shown in Appendix are used. The Bj , Cj and Fj are some expressions
of material consistents, Eij and λij (i = 1, 2, 3, j = 1, . . . , 4). Then the
solution of the unknown functions can be obtained. Substitute the solution
of the above equation and using the conditions of the crack surface, after
considering the asymptotic when s → ∞ and α → 0, the following equation
can be obtained
¸
Z ·
Im(ω1 )
1 b
−
+h1 (u, x)+2K2s +2h2 (u, x) g(u)du = −σ0 (x)e−δx , (14)
π a
u−x
the Eq. (14) is the first kind of Fredholm integral equation, which can be
solved by the method of [15].
The stress intensity factors of the internal crack tips can be express as
q
KI (a) = lim
x→a
2 (a − x) σyy (x, 0) = −Im (ω11 ) eδa
N
X
n=1
an ,
N
q
X
(−1)n an .
gI (b) = lim 2 (x−b) σyy (x, 0) = −Im (ω11 ) eδb
x→b
(15)
n=1
√
For convenience, the SIFs are normalized by k0 = σ0 a0 h, where σ0 is
uniform crack surface pressure.
3.
Numerical Results and Discussion
In the following analysis, we will study the influence of the length of the
crack, the location of the crack, the functionally graded strip’s width h and
the gradient parameters of functionally graded material δ on the normalized
stress intensity factors (SIFs) of the crack.
Firstly, the influence of the length of the crack and the gradient parameters of functionally graded material δ on the normalized stress intensity
factors (SIFs) of the crack will be discussed. Here h = 5.0, δh = −1, 0, 1, 2,
c = 0.4h. It can be found in the Fig.2–Fig.3 that the normalized intensity
factors of both crack tips KI (a) and KI (b) increases with the increase of the
normalized half crack length a0 and the gradient parameters of functionally
graded material δ. Therefore, to prevent the coating crack from growing
toward the interface, the gradient parameters of functionally graded material δ should be chosen as δ < 0.
295
Mode I Crack Problem for a Substrate-Coating Structure
0
0
δ*h=−1
δ*h=0
δ*h=1
δ*h=2
−0.1
−0.2
−0.2
K1(a)/K0
−0.3
−0.4
−0.4
−0.5
−0.5
−0.6
−0.6
−0.7
−0.7
0.1
0.15
0.2
0.25
0.3
0.35
−0.8
0.05
0.4
0.1
0.15
0.2
a0/h
0.25
0.3
0.35
0.4
a0/h
Figure 2:
The normalized stress intena
sity factor KI (b) versus h0
Figure 3:
The normalized stress intena
sity factor KI (a)versus h0
Secondly, we will discuss the influence of the location of the crack on the
normalized stress intensity factors (SIFs) of the crack. Here δh = 1, c =
0.5, 1, 1.5, 2. It can be found from Fig.4–Fig.5 that the normalized intensity
factors of both crack tips KI (a) and KI (b) increases with the increase of
the c = b+a
2 .
0
0.1
c=0.5
c=1
c=1.5
c=2
0
−0.1
c=0.5
c=1
c=1.5
c=2
−0.1
−0.2
−0.2
−0.3
K1(a)/K0
−0.3
K1(b)/K0
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K1(b)/K0
−0.3
−0.8
0.05
δ*h=−1
δ*h=0
δ*h=1
δ*h=2
−0.1
−0.4
−0.4
−0.5
−0.5
−0.6
−0.6
−0.7
−0.7
−0.8
−0.8
−0.9
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
−0.9
0.05
0.1
a0
Figure 4:
The normalized stress intensity factor KI (b)versus a0
0.15
0.2
0.25
0.3
0.35
0.4
a0
Figure 5:
The normalized stress intensity factor KI (a)versus a0
Finally, we will discuss the influence of the width of the functionally
graded orthotropic strip on the normalized stress intensity factors (SIFs) of
the crack. Here h = 1, 2, 3, 4, δh = 1, c = 0.5. It can be found from
the Fig.6–Fig.7 that the normalized intensity factors of both crack tips
KI (a) and KI (b) decreases with the increase of the h, but as the increase
of the strip width and the increase of the crack length, the effect is not
obvious, so increasing the width of coating is not an effective way to restrain
the expansion of the crack.
296
Li-Fang Guo, Xing Li and Sheng-Hu Ding
0.9
0.9
h=1
h=2
h=3
h=4
0.8
0.7
h=1
h=2
h=3
h=4
0.8
0.7
0.6
0.6
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K1(b)/K0
K1(a)/K0
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0.05
0
0.1
0.15
0.2
0.25
0.3
0.35
0.4
a0
Figure 6:
The normalized stress intensity factor KI (b)versus a0
4.
−0.1
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
a0
Figure 7: The normalized stress intensity factor KI (a)versus a0 for
Conclusion
In this paper, the mode I crack problem for a functionally graded orthotropic strip bonded to orthotropic substrate is studied analytically. The
influences of the nonhomogeneity constants and geometric parameters on
the stress intensity factors are investigated. The result may be help for the
analysis and design of functionally graded orthotropic coating-substrate
structures.
References
[1] F. Delale and F. Erdogan, On the mechanical modeling of the interfacial region in
bonded half-planes, International Journal of Applied Mechanic 55 (1988), 317–324.
[2] S. Dag and F. Erdogan, A surface crack in a graded medium loaded by a sliding rigid
stamp, Engineering Fracture Mechanics 69 (2002), 1729–1751.
[3] Y. D. Li, W. Tan and H. C. Zhang, Anti-plane transient fracture analysis of the
functionally gradient elastic bi-material weak/infinitesimal-discontinuous interface,
International Journal of fracture 142 (2006), 163–171.
[4] L. C. Guo, L. Z. Wu and L. Ma, The interface crack problem under a concentrated load
for a functionally graded coating- substrate composite system, Composite structure
63 (2004), 397–406.
[5] L. C. Guo, L. Z. Wu, L. Ma and T. Zeng, Fracture analysis of a functionally graded
coating-substrate structure with a crack perpendicular to the interface, Part I: static
problem, International Journal of fracture 127 (2004), 21–38.
[6] M. Ozturk and F. Erdogan, Mode-I crack problem in an inhomogeneous orthotropic
medium, Journal of Engineering Science 35 (1997), 869C-883.
[7] S. Dag, B. Yildirim and F. Erdogan, Interface crack problems in graded orthotropic
media: Analytical and computational approaches. International Journal of Fracture
130 (2004), 471C-496.
[8] L. C. Guo, L. Z. Wu, T. Zeng and L. Ma, Mode I crack problem for a functionally
graded orthotropic strip, European Journal of Mechanics A/solids 23 (2004), 219–
234.
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[9] H. M. Xu, X. F. Yao, X. Q. Feng and H. Y. Yeh, Fundamental solution of a power-law
orthotropic and half-space functionally graded material under line loads, Composites
Science and Technology 68 (2008), 27–34.
[10] Vijaya Bhaskar Chalivendra, Mode-I crack-tip stress fields for inhomogeneous orthotropic medium, Mechanics of Materials 40 (2008), 293–301.
[11] J. H. Kim and G. H. Paulino, Mixed-mode fracture of orthotropic functionally graded
materials using finite elements and the modified crack closure method, Engineer
fracture mechanic 69 (2002), 1557–1586.
[12] J. H. Kim and G. H. Paulino, Mixed-mode J-integral formulation and implementation
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materials, mechanic material 35 (2003), 107–128.
[13] A. Kawasaki and R. Watanabe, Thermal fracture behavior of metal/ceramic functionally graded materials, Engineering Fracture Mechanics 69 (2000), 1713–1728.
[14] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Elsevier/Academic Press, Amsterdam, 2007.
[15] X. Li, Integral equations, Science Press, Beijing, 2008.
Appendix

£
¤ 
c660 (s + iδ) s − c220 λ21j i 




, 
E1j =




[c
s
+
c
(s
+
iδ)]
λ


120
660
1j






2
c660 λ2j (λ2j + δ) − α c220
j = 1 . . . , 4,
E2j =
,

α [c120 λ2j + c660 (λ2j + δ)] 










c660 λ23j − α2 c220




E
=
,
 3j

α (c120 + c660 ) λ3j
r


q
Ω11 1




2


±
Ω11 −4Ω12 ,
λ1j = − −




2
2




q


p


2
2
−δ ± δ −2Ω21 ±2 Ω21 −4Ω22
j = 1 . . . , 4,
λ2j =
,



2




√ r


q




2


2
 λ3j =

−Ω31 ± Ω31 − 4Ω32 ,
2
(c2120 −c120 c220 )s(s+iδ)+(2s2 +2isδ−δ 2 )c120 c660
,
c220 c660
c110 s2 (s+iδ)2
Ω12 =
,
c220
£ 2
¤
µ
¶
c120 −c110 c220 +2c120 c660 α2
c220 2 c120 2 2
Ω21 =
, Ω22 =
α +
δ α ,
c110 c660
c110
c110
¸
· 2
c120
c220
c220 4
c120
+2
−
α2 , Ω32 =
α ,
Ω31 =
c110 c660
c110
c660
c110
(A1)
(A2)
Ω11 =
(A3)
(A4)
(A5)
298
Li-Fang Guo, Xing Li and Sheng-Hu Ding
Z
∞
Boundary Value Problems, Integral Equations and Related Problems Downloaded from www.worldscientific.com
by NATIONAL UNIVERSITY OF SINGAPORE on 10/25/17. For personal use only.
1
(q cos λ + p sin λ) ,
2 + q2
p
Z0 ∞
p > 0,
1
−px
e
cos (qx + λ) dx = 2
(p cos λ − q sin λ) ,
p + q2
0
e−px sin (qx + λ) dx =
¢
¢¡
¡
ic660 c120 + c220 p211 c120 + c220 p212
,
ω1 =
c220 (c120 + c660 ) p11 p12 (p11 + p12 )
µ 4
¶
Z
1 ∞ X
h1 (u, x) =
C1j q1j − ω1 eis(u−x) ds,
2 −∞ j=3
·
K2s =
¸
Λ1
Λ2
Λ3
Λ4
+
+
+
,
p21 (x + u) p21 x + p22 u p22 x + p21 u p22 (x + u)
4
∞· X
Z
h2 (u, x) =
0
(A6)
(A7)
(A8)
C2j eλ2j x A2j −Λ1 e−p21 α(x+u)
j=3
¸
−α(p21 x+p22 u)
−α(p22 x+p21 u)
−p22 α(x+u)
−Λ2 e
−Λ3 e
−Λ4 e
dα,
¢¡
¢
1¡
Λ1 = c120 p221 + c220 c120 c660 +c2120 −c110 c220 +c110 c660 p222
£ 4
¡
¢
¤
× −c110 c660 p421 + c110 c220 − c2120 − 2c120 c660 p221 − c220 c660
1
×
2
2,
2
2
c660 (c660 + c120 ) c110 p21 (p21 + p22 ) (p21 − p22 )
1
(c120 p221 + c220 )
4
×[−c110 c660 p422 +(c110 c220 −c2120 −2c120 c660 )p222 −c220 c660 ]
¡
¢
× c110 c220 − c110 c660 p21 p22 − c2120 − c120 c660
(A9)
(A10)
Λ2 =
×
1
c660 c2110
(c660 +
c120 ) p222
2
(A11)
2,
(p21 + p22 ) (p21 − p22 )
¢
1¡
c120 p222 + c220
4
×[−c110 c660 p421 +(c110 c220 −c2120 −2c120 c660 )p221 −c220 c660 ]
¡
¢
× c110 c220 − c110 c660 p21 p22 − c2120 − c120 c660
Λ3 =
×
1
2
2,
c660 c2110 (c660 + c120 ) p221 (p21 + p22 ) (p21 − p22 )
(A12)
Mode I Crack Problem for a Substrate-Coating Structure
¢
1¡
c120 p222 + c220
4
¢
¡
×[−c110 c660 p422 + c110 c220 −c2120 −2c120 c660 p222 −c220 c660 ]
¢
¡
× −c120 c660 − c2120 + c110 c220 − c110 c660 p221
299
Boundary Value Problems, Integral Equations and Related Problems Downloaded from www.worldscientific.com
by NATIONAL UNIVERSITY OF SINGAPORE on 10/25/17. For personal use only.
Λ4 = −
×
(A13)
1
2
2,
(p21 + p22 ) (p21 − p22 )
√
h p
i
2 p
p1j = −
ξ11 aa1 ±bb1 , ξ1j = sign Re aa1 ± bb1 ,
√2
h p
i j = 1, 2,
2 p
p2j =
ξ21 aa2 ± bb2 , ξ21 = sign Re aa2 ± bb2 ,
2
c110 c220 − c120 (c120 + 2c660 )
aa1 =
,
c220 c660
q
c660 c2110
(c660 +
c120 ) p222
(A14)
2
bb1 =
(c2120 − c110 c220 ) [(c120 + 2c660 ) − c110 c220 ]
c220 c660
c110 c220 − c2120 − 2c120 c660
,
aa2 =
c110 c660
q
bb2 =
(A15)
2
(c2120 − c110 c220 ) [(c120 + 2c660 ) − c110 c220 ]
c110 c660
,
.
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